Computer Graphics. - Introduction to Ray Tracing - Philipp Slusallek

Transcription

1 Computer Graphics - Introduction to Ray Tracing - Philipp Slusallek

2 Rendering Algorithms Rendering Definition: Given a 3D scene as input and a camera, generate a 2D image as a view from the camera of the 3D scene Algorithms Ray Tracing Declarative scene description Physically-based simulation of light transport Rasterization Traditional procedural/imperative drawing of a scene content

3 Scene Surface Geometry 3D geometry of objects in a scene Geometric primitives triangles, polygons, spheres, Surface Appearance Color, texture, absorption, reflection, refraction, subsurface scattering Mirror, glass, glossy, diffuse, Illumination Position and emission characteristics of light sources Note: Light is reflected off of surfaces! Secondary/indirect/global illumination Assumption: air/empty space is totally transparent Simplification that excludes scattering effects in participating media volumes Later also volume objects, e.g. smoke, solid object (CT scan), Camera View point, viewing direction, field of view, resolution,

4 OVERVIEW OF RAY-TRACING

5 Ray Tracing Can Produce Realistic Images By simulating light transport

6 Light Transport (1)

7 Light Transport (2) Light Distribution in a Scene Dynamic equilibrium Forward Light Transport Shoot photons from the light sources into scene Reflect at surfaces and record when a detector is hit Photons that hit the camera produce the final image Most photons will not reach the camera Particle Tracing Backward Light Transport Start at the detector (camera) Trace only paths that might transport light towards it May try to connect to occluded light sources Ray Tracing

8 Ray Tracing Is Fundamental rendering algorithm Automatic, simple and intuitive Easy to understand and implement Delivers correct images by default Powerful and efficient Many optical global effects Perspective Machine, Albrecht Dürer Shadows, reflections, refractions, Efficient real-time implementation in SW and HW Can work in parallel and distributed environments Logarithmic scalability with scene size: O(log n) vs. O(n) Output sensitive and demand driven Concept of light rays is not new Empedocles ( BC), Renaissance (Dürer, 1525), Uses in lens design, geometric optics,

9 Fundamental Ray Tracing Steps Generation of primary rays Rays from viewpoint along viewing directions into 3D scene (At least) one ray per picture element (pixel) Ray casting Traversal of spatial index structures Ray-primitive intersection Shading the hit point Determine pixel color Energy (color) travelling along primary ray Needed Local material color, object texture and reflection properties Local illumination at intersection point Compute through recursive tracing of rays Can be hard to determine correctly

10 Ray Tracing Pipeline (1) Ray Generation Ray Traversal Intersection Shading Pixel Color

11 Ray Tracing Pipeline (2) Ray Generation Ray Traversal Intersection Shading Pixel Color

12 Ray Tracing Pipeline (3) Ray Generation Ray Traversal Intersection Shading Pixel Color

13 Ray Tracing Pipeline (4) Ray Generation Ray Traversal Intersection Shading Pixel Color

14 Ray Tracing Pipeline (5) Ray Generation Ray Traversal Intersection Shading Pixel Color

15 Ray Tracing Pipeline (6) Ray Generation Ray Traversal Intersection Shading Pixel Color

16 Ray Tracing Pipeline (7) Ray Generation Ray Traversal Intersection Shading Pixel Color

17 Recursive Ray Tracing reflected ray primary ray pixel light source shadow rays refracted ray image plane Searching recursively for paths to light sources Interaction of light & material at intersections Recursively trace new rays in reflection, refraction, and light direction Reflected Eye Cube Refracted viewpoint Sphere Cylinder

18 Ray Tracing Algorithm Trace(ray) Search the next intersection point (hit, material) Return Shade(ray, hit, material) Shade(ray, hit, material) For each light source if ShadowTrace(ray to light source, distance to light) Calculate reflected radiance (i.e. Phong) Adding to the reflected radiance If mirroring material Calculate radiance in reflected direction: Trace(R(ray, hit)) Adding mirroring part to the reflected radiance Same for transmission Return reflected radiance ShadowTrace(ray, dist) Return false, if intersection with distance < dist has been found Can be changed to handle transparent objects as well But not with refraction

19 Shading Intersection point determines primary ray s color Diffuse object: color at intersection point No variation with viewing angle: diffuse (Lambertian) Perfect reflection/refraction (mirror, glass) Only one outgoing direction Trace one secondary ray Non-Lambertian Reflectance Appearance depends on illumination and viewing direction Local Bi-directional Reflectance Distribution Function (BRDF) Illumination Point/directional light sources Area light sources Approximate with multiple samples / shadow rays Indirect illumination See Realistic Image Synthesis (RIS) course in next semester More details later

20 Common Approximations Usually RGB color model instead of full spectrum Finite # of point lights instead of full indirect light Approximate material reflectance properties Ambient: constant, non-directional background light Diffuse: light reflected uniformly in all directions Specular: perfect reflection, refraction Reflection models are often empirical

21 Ray Tracing Features Incorporates into a single framework Hidden surface removal Front to back traversal Early termination once first hit point is found Shadow computation Shadow rays/ shadow feelers are traced between a point on a surface and a light sources Exact simulation of some light paths Reflection (reflected rays at a mirror surface) Refraction (refracted rays at a transparent surface, Snell s law) Limitations Many reflections (exponential increase in number of rays) Indirect illumination requires many rays to sample all incoming directions Easily gets inefficient for full global illumination computations Solved with Path Tracing ( later)

22 Ray Tracing Can Produce Realistic Images By simulating light transport

23 What is Possible? Models Physics of Global Light Transport Dependable, physically-correct visualization

24 VW Visualization Center

25 Realistic Visualization: CAD

26 Realistic Visualization: VR/AR

27 Lighting Simulation

28 What is Possible? Huge Models Logarithmic scaling in scene size 12.5 Million Triangles ~1 Billion Triangles

29 Outdoor Environments 90 x 10^12 (trillion) triangles

30 Boeing 777 Boeing 777: ~350 million individual polygons, ~30 GB on disk

31 Volume Visualization Iso-surface rendering

32 Games?

33 Ray Tracing in CG In the Past Only used as an off-line technique Was computationally far too demanding (minutes to hours per frame) Believed to not be suitable for a HW implementation More Recently Interactive ray tracing on supercomputers [Parker, U. Utah 98] Interactive ray tracing on PCs [Wald 01] Distributed Real-time ray tracing on PC clusters [Wald 01] RPU: First full HW implementation [Siggraph 2005] Commercial tools: Embree/OSPRey (Intel/CPU), OptiX (Nvidia/GPU) Complete film industry has switched to ray tracing (Monte-Carlo) Own conference Symposium on Interactive RT, now High-Performance Graphics (HPG) Ray tracing systems Research: PBRT (offline, physically-based, based on book, OSS), Mitsuba renderer (EPFL), imbatracer (SB), Commercial: V-Ray (Chaos Group), Corona (Render Legion), VRED (Autodesk), MentalRay/iRay (MI),

34 Ray Casting Outside CG Tracing/Casting a ray Type of query Is there a primitive along a ray How far is the closest primitive Other uses than rendering Volume computation Sound waves tracing Collision detection

35 RAY-PRIMITIVE INTERSECTIONS

36 Basic Math - Ray Ray parameterization r t = o + t d, t R; o, d R 3 : origin and direction Ray All points on the graph of r t, with t R 0+ t=3 o d t=1 t=2

37 Pinhole Camera Model // For given image resolution {resx, resy} // Loop over pixel raster coordinates [0, res-1] for(prcx = 0; prcx < resx; prcx++) for(prcy = 0; prcy < resy; prcy++) { // Normalized device coordinates [0, 1] ndcx = (prcx + 0.5) / resx; ndcy = (prcy + 0.5) / resy; // Screen space coordinates [-1, 1] sscx = ndcx * 2-1; sscy = ndcy * 2-1; // Generate direction through pixel center d = f + sscx x + sscy y; d = d / d ; // May normalize here // Trace ray and assign color to pixel color = trace_ray(o, d); write_pixel(prcx, prcy, color); d } Image plane y x spanning vectors u up-vector f focal vector o origin, POV

38 Basic Math - Sphere Sphere S c R 3, r R: center and radius p R 3 : p S p c p c r 2 = 0 The distance between the points on the sphere and its center equals the radius p 1 p 1 - c c p 2 - c p 2

39 Ray-Sphere Intersection Given Ray: r t = o + t d, t R; o, d R 3 Sphere: c R 3, r R: p R 3 : p S p c p c r 2 = 0 Find closest intersection point Algebraic approach: substitute ray equation p c p c r 2 = 0 with p = o + t d t 2 d d + 2t d o c + o c ( o c) r 2 = 0 Solve for t

40 Ray-Sphere Intersection (2) Given Ray: r t = o + t d, t R; o, d R 3 Sphere: c R 3, r R: p R 3 : p S p c p c r 2 = 0 Find closest intersection point Geometric approach Ray and center span a plane Solve in 2D Compute b o, b c OBC = 90 Intersection(s) if b c r r c Be aware of floating point issues if o is far from sphere o d o - c

41 Basic Math - Plane Plane P n, a R 3 : normal and point in P p R 3 : p P p a n = 0 The difference vector between any two points on the plane is either 0 or orthogonal to the plane s normal n a p 1 - a p 2 p 2 - a n p 1 a

42 Ray-Plane Intersection Given Ray: r t = o + t d, t R; o, d R 3 Plane: n, a R 3 : normal and point in P Compute intersection point Plane equation: p P p a n = 0 p n D = 0, with D = a n Substitute ray parameterization: ( o + t d) n D = 0 Solve for t 0,1 or infinitely many solutions

43 Ray-Disc Intersection Intersect ray with plane Discard intersection if p a > r

44 Basic Math - Triangle Triangle T a, b, c R 3 : vertices Affine combinations of a, b, c points in the plane Non-negative coefficients that sum up to 1 points in the triangle p R 3 : p T λ 1,2,3 R 0+, λ 1 + λ 2 + λ 3 = 1 and p = λ 1 a + λ 2 b + λ 3 c c Barycentric coordinates λ 1,2,3 λ 1 = S pbc /S abc S: signed area of triangles λ 2 λ 3 p λ 1 a b

45 Barycentric Coordinates Triangle T a, b, c R 3 : vertices λ 1,2,3 : barycentric coordinates λ 1 + λ 2 + λ 3 = 1 λ 1 = S pbc /S abc, etc. c (0, 0, 1) λ 3 = 1 2 ( 1 2, 0, 1 2 ) (0, 1 2, 1 2 ) λ 3 = 1 3 ( 1 3,1 3,1 3 ) a (1,0,0) (0, 1, 0) ( 1, 1, 0) b 2 2

46 Triangle Intersection: Plane-Based Compute intersection with triangle plane Compute barycentric coordinates Signed areas of subtriangles Can be done in 2D, after projection onto major plane, depending on largest normal vector component Test for positive BCs z o n d y x

47 Triangle Intersection Edge-Based (1) 3D linear function across triangle (3D edge functions) Ray: o + t d, t R; o, d R 3 Triangle: a, b, c R 3 c a b o d

48 Triangle Intersection Edge-Based (2) 3D linear function across triangle (3D edge functions) Ray: o + t d, t R; o, d R 3 Triangle: a, b, c R 3 n ab = b o (a o) n ab is the signed area of OAB (2 times) c n ab a b o d

49 Triangle Intersection Edge-Based (3) 3D linear function across triangle (3D edge functions) Ray: o + t d, t R; o, d R 3 Triangle: a, b, c R 3 n ab = b o (a o) n ab is the signed area of OAB (2 times) λ 3 (t) = n ab t d Volume of OABP (6 times) For t = t hit n ab p c a b o d

50 Triangle Intersection Edge-Based (4) 3D linear function across triangle (3D edge functions) Ray: o + t d, t R; o, d R 3 Triangle: a, b, c R 3 n ab = b o (a o) n ab is the signed area of OAB (2 times) λ 3 (t) = n ab t d Volume of OABP (6 times) For t = t hit λ 1,2 (t) = n bc,ac t d Normalize λ i = λ i (t) λ 1 (t)+λ 2 (t)+λ 3 (t), i = 1,2,3 Length of t d cancels out o n ab d a p c b

51 Triangle Intersection Edge-Based (5) 3D linear function across triangle (3D edge functions) Ray: o + t d, t R; o, d R 3 Triangle: a, b, c R 3 n ab = b o (a o) n ab is the signed area of OAB (2 times) λ 3 (t) = n ab t d Volume of OABP (6 times) For t = t hit λ 1,2 (t) = n bc,ac t d Normalize λ i = λ i (t) λ 1 (t)+λ 2 (t)+λ 3 (t), i = 1,2,3 n ab a p c b For positive BCs Compute p = λ 1 a + λ 2 b + λ 3 c o d

52 Quadrics Implicit f(x, y, z) = v Ray equation x = xo + t xd y = yo + t yd z = zo + t zd Solve for t Assignment

53 Axis Aligned Bounding Box Given Ray: o + t d, t R; o, d R 3 Axis aligned bounding box (AABB): p min, p max R 3 Bounded Volume p max p min

54 Ray-Box Intersection Given Ray: o + t d, t R; o, d R 3 Axis aligned bounding box (AABB): p min, p max R 3 Slabs test for ray-box intersection Ray enters the box in all dimensions before exiting in any max({t i near i = x, y, z}) < min({t i far i = x, y, z}) t x near o Bounded Volume t y near (smaller) t x far (larger) t x near o Bounded Volume t x far (smaller) t y near (larger) t y far t y far

55 History of Intersection Algorithms Ray-geometry intersection algorithms Polygons: [Appel 68] Quadrics, CSG: [Goldstein & Nagel 71] Recursive Ray Tracing: [Whitted 79] Tori: [Roth 82] Bicubic patches: [Whitted 80, Kajiya 82] Algebraic surfaces: [Hanrahan 82] Swept surfaces: [Kajiya 83, van Wijk 84] Fractals: [Kajiya 83] Deformations: [Barr 86] NURBS: [Stürzlinger 98] Subdivision surfaces: [Kobbelt et al 98]

56 Precision Problems Cause of surface acne